First we prove the assertion for \(\mathcal {G}=\mathcal {F}.\)
Let \((A_m,u_m)\in \mathcal {C}_m\)  be a minimizer of \(\mathcal {F}\)  satisfying the volume constraint \(|A_m|=\mathtt {v}\)  and
let \((A_{m_h},u_{m_h}),\)  and \(A\)  and \(u\) 
be as in Proposition REF .
By (REF ), (REF ) and (REF ) we have
\(\inf \limits _{(B,v)\in \mathcal {C},\,|B|=\mathtt {v}} \mathcal {F}(B,v) = \lim \limits _{h\rightarrow +\infty } \mathcal {F}(A_{m_h},u_{m_h}) \ge \widetilde{\mathcal {F}}(\mathrm {Int}(A),u).\) 
Since \( |\mathrm {Int}(A)|=\mathtt {v},\)  by Propositions REF  and  REF
\(\inf \limits _{(B,v)\in \mathcal {C},\,|B|=\mathtt {v}} \mathcal {F}(B,v)=\inf \limits _{(B,v)\in \widetilde{\mathcal {C}},\,|B|=\mathtt {v}} \widetilde{\mathcal {F}}^{\lambda _0}(B,v)=\widetilde{\mathcal {F}}^{\lambda _0}(\mathrm {Int}(A),u)=\widetilde{\mathcal {F}}(\mathrm {Int}(A),u),\) 
hence, \((\mathrm {Int}(A),u)\)  is a minimizer of \(\widetilde{\mathcal {F}}^{\lambda _0}\)  in \(\widetilde{\mathcal {C}}.\)  Then by Proposition REF  there exists \((A^{\prime },u^{\prime })\in \mathcal {C}\)  such that
\(\widetilde{\mathcal {F}}(\mathrm {Int}(A),u) = \mathcal {F}(A^{\prime },u^{\prime }),\) 
and hence, in view of (REF ), \((A^{\prime },u^{\prime })\)  is a solution to (REF ).
The proof of the second assertion (i.e., the existence of \(\lambda _1\)  for which the set of minimizers in \(\mathcal {C}\)  of both \(\mathcal {F}\)  and \(\mathcal {F}^\lambda \)  coincide for all \(\lambda \ge \lambda _1\) ) can be done using the first one and also following the arguments of [1]} and [2]}. Without loss of generality we assume that \(\lambda _1\ge \lambda _0,\)  where \(\lambda _0\)  is given by Proposition REF .
Now we prove Theorem REF  for \(\mathcal {G}=\widetilde{\mathcal {F}}.\)  We have already shown above that the configuration \((\mathrm {Int}(A),u)\)  given by Proposition REF  solves the minimum problem (REF ) with \(\mathcal {G}=\widetilde{\mathcal {F}}.\)  In view of (REF ) every volume-constraint minimizer of \(\widetilde{\mathcal {F}}\)  also minimizer of \(\widetilde{\mathcal {F}}^\lambda \)  for all \(\lambda \ge \lambda _1.\)  To prove the converse assertion, we fix any minimizer \((A,u)\in \widetilde{\mathcal {C}}\)  of \(\widetilde{\mathcal {F}}^\lambda \)  for \(\lambda \ge \lambda _1.\)  By Proposition REF  there exists \((A^{\prime },u^{\prime })\in \mathcal {C}\)  such that \(|A^{\prime }|=|A|\)  and \(\mathcal {F}(A^{\prime },u^{\prime })=\widetilde{\mathcal {F}}(A,u).\)  By the first part of the proof and (REF ) we know that
\(\inf \limits _{(B,v)\in \mathcal {C}}\mathcal {F}^\lambda (B,v) = \inf \limits _{(B,v)\in \mathcal {C},|B|=\mathtt {v}}\mathcal {F}(B,v) = \inf \limits _{(B,v)\in \widetilde{\mathcal {C}}}\widetilde{\mathcal {F}}^\lambda (B,v) =\mathcal {F}^\lambda (A^{\prime },u^{\prime }).\) 
Hence, \((A^{\prime },u^{\prime })\)  is the minimizer of \(\mathcal {F}^\lambda .\)  Since \(\lambda \ge \lambda _1\)  according to the first part of the proof, \( |A^{\prime }|=\mathtt {v}.\)  Hence, \(|A|=\mathtt {v}\)  and \((A,u)\)  minimizer of (REF ).
